ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

10. Partial Differential Equations

1991 MS Classification: 35,46 Pulkina, Ludmila, Samara State University, Russia A nonlocal problem for the hyperbolic equation Certain problems, arising in modern physics, can be reduced to nonlocal problems for the second order partial differential equations. Consider the problem for the hyperbolic equation

Lu ≡ u_{xx} + (Au)_{x} + (Bu)_{y} + Cu = f(x, y)

in the domain D = {(x, y) : 0 < x < a, 0 < y < b} with the following conditions: ∫_{0}^{a} u(x, y)dx = ∫_{0}^{b} u(x, y)dy = 0. It is to be note that if integral conditions are not homogeneous we can easily reduce them to homogeneous by constructing a function satisfying the same integral conditions. We define the function space L̃_{2} = {u : u ∈ L_{2}(D), ∫_{0}^{a} udx = ∫_{0}^{b} udy = 0}. The weak solution of the posed problem is a function u(x, y) ∈ L̃_{2}(D) such that for every v(x, y) ∈ L̃_{2}(D), (u, L^{∗}lv)_{L_{2}} = (f, lv)_{L_{2}}, where lv ≡ ∫_{0}^{y} ∫_{0}^{x} v(t, τ)dtdτ. The existence and uniqueness of weak solution are proved. In this end of view two sided a priori estimate is derived. To show the existence of the solution the Galerkin method is used.